Logarithmic integral function: Difference between revisions

Revision as of 00:57, 1 December 2002

Revision as of 17:17, 27 January 2003

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In some 'esoteric' areas of [[mathematics]], the '''logarithmic integral''' or '''integral logarithm''' li(''x'') is a [[function|non-elementary [[function]] defined for all positive [[real number]]s ''x''≠ 1 by the definite [[integral|definite integral]]:

:<math> {\rm li} (x) = \int_{0}^{x} {1\over \ln t} dt \; . </math>

:li(''x'') = &int;₀^{^''x''} 1/ln ''t'' d''t''.

Here, ln denotes the [[natural logarithm]]. The function 1/ln ''t'' has a [[mathematical singularity|singularity]] at ''t'' = 1, and the integral for ''x'' > 1 has to be interpreted as ''Cauchy's principal value'':

as ''Cauchy's principal value'':

:<math> {\rm li} (x) = \lim_{\varepsilon \to 0} \left( \int_{0}^{1-\varepsilon} {1\over \ln t} dt + \int_{1+\varepsilon}^{x} {1\over \ln t} dt \right) \; . </math>

:li(''x'') = lim_{&epsilon;&rarr;0} ( &int;₀^{^1-&epsilon;} 1/ln ''t'' d''t'' + &int;_1+&epsilon;^{^x} 1/ln ''t'' d''t'' ).

The growth behavior of this function for ''x'' &rarr; &infin; is

:li(''x'') = &Theta;(''x''/ln(''x''))

:<math> {\rm li} (x) = \Theta \left( {x\over \ln x} \right) \; . </math>

(see [[big O notation]]).

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:&pi;(''x'') ~ Li(''x'')

where &pi;(''x'') denotes a [[multiplicative function]] - the number of primes smaller than or equal to ''x'', and Li(''x'') is the [[offset logarithmic integral]] function, related to li(''x'') by Li(''x'') = li(''x'') - li(2).

The offset logarithmic integral gives a slightly better estimate to the &pi; function than li(''x''). The function li(''x'') is related to the ''[[exponential integral]]'' Ei(''x'') via the equation

The function li(''x'') is related to the ''[[exponential integral]]'' Ei(''x'') via the equation

:li(''x'') = Ei (ln ''x'') &nbsp;&nbsp; for all positive real ''x'' &ne; 1.

This leads to series expansions of li(''x''), for instance:

:li(e^''u'') = &gamma; + ln |''u''| + <font size="+1"> &sum;</font>_n=1^&infin; ''u''ⁿ/(n &middot; n!) for ''u'' &ne; 0

where &gamma; &asymp; 0.57721 56649 01532 ... is the [[Euler-Mascheroni gamma constant]].

:<math> {\rm li} (e^{u}) = \gamma + \ln \left| u \right| + \sum_{n=1}^{\infty} {u^{n}\over n \cdot n!} \quad {\rm for} \; u \ne 0 \; , </math>

The function li(''x'') has a single positive zero; it occurs at ''x'' &asymp; 1.45136 92348 ...; this number is known as the ''[[Ramanujan]]-Soldner constant''.

where &gamma; &asymp; 0.57721 56649 01532 ... is the [[Euler-Mascheroni gamma constant]]. The function li(''x'') has a single positive zero; it occurs at ''x'' &asymp; 1.45136 92348 ...; this number is known as the ''[[Ramanujan]]-Soldner constant''.